(a) Prove that the above definition makes sense, by showing that the series converges for every complex number z. Moreover, show that the conver- gence is uniform5 on every bounded subset of C. (b) If 21, 22 are two complex numbers, prove that e²¹ e²² = e²¹+22. [Hint: Use the binomial theorem to expand (z₁+z2)", as well as the formula for the binomial coefficients.]
(a) Prove that the above definition makes sense, by showing that the series converges for every complex number z. Moreover, show that the conver- gence is uniform5 on every bounded subset of C. (b) If 21, 22 are two complex numbers, prove that e²¹ e²² = e²¹+22. [Hint: Use the binomial theorem to expand (z₁+z2)", as well as the formula for the binomial coefficients.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
DO NOT COPY OTHER ANSWERS. THEY ARE INCORRECT. Please answer ALL PARTS and show your work!
![(c) Show that if z is purely imaginary, that is, z = iy with y R, then
ey cos y + i sin y.
This is Euler's identity. [Hint: Use power series.]
(d) More generally,
whenever x, y € R, and show that
ex+iye (cosy + i sin y)
(e) Prove that e² = 1 if and only if z = 2πki for some integer k.
(f) Show that every complex number z = x+iy can be written in the form
z = re²0
where r is unique and in the range 0 ≤r <∞, and 0 € R is unique up to
an integer multiple of 27. Check that
r = |z| and
whenever these formulas make sense.
(g) In particular, i = eiñ/2. What is the geometric meaning of multiplying a
complex number by i? Or by e for any 0 € R?
(h) Given R, show that
eio
cos=
|e+iy| = e.
and then show that
+e-io
2
These are also called Euler's identities.
(i) Use the complex exponential to derive trigonometric identities such as
cos(+9)= cos cos - sin sin 9,
0 = arctan(y/x)
2 sin sin
=
2 sin cos =
- e-io
2i
and sin 0 =
cos (0) cos(0 + y),
sin(0+ p) + sin(0-4).
This calculation connects the solution given by d'Alembert in terms of
traveling waves and the solution in terms of superposition of standing
waves.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90b5da61-7c9e-41cf-88f9-128e7ad2af51%2F290b67e6-bebc-47a3-9b4a-216c0dbb6a6a%2Frgvdcfg_processed.png&w=3840&q=75)
Transcribed Image Text:(c) Show that if z is purely imaginary, that is, z = iy with y R, then
ey cos y + i sin y.
This is Euler's identity. [Hint: Use power series.]
(d) More generally,
whenever x, y € R, and show that
ex+iye (cosy + i sin y)
(e) Prove that e² = 1 if and only if z = 2πki for some integer k.
(f) Show that every complex number z = x+iy can be written in the form
z = re²0
where r is unique and in the range 0 ≤r <∞, and 0 € R is unique up to
an integer multiple of 27. Check that
r = |z| and
whenever these formulas make sense.
(g) In particular, i = eiñ/2. What is the geometric meaning of multiplying a
complex number by i? Or by e for any 0 € R?
(h) Given R, show that
eio
cos=
|e+iy| = e.
and then show that
+e-io
2
These are also called Euler's identities.
(i) Use the complex exponential to derive trigonometric identities such as
cos(+9)= cos cos - sin sin 9,
0 = arctan(y/x)
2 sin sin
=
2 sin cos =
- e-io
2i
and sin 0 =
cos (0) cos(0 + y),
sin(0+ p) + sin(0-4).
This calculation connects the solution given by d'Alembert in terms of
traveling waves and the solution in terms of superposition of standing
waves.
![4. For z € C, we define the complex exponential by
e²=
=
Σ
n=0
n!
(a) Prove that the above definition makes sense, by showing that the series
converges for every complex number z. Moreover, show that the conver-
gence is uniform5 on every bounded subset of C.
(b) If 2₁, 22 are two complex numbers, prove that e²¹ e²² = e²¹+²². [Hint: Use
the binomial theorem to expand (z₁+z2)”, as well as the formula for the
binomial coefficients.]
5 A sequence of functions {fn(z)}_₁ is said to be uniformly convergent on a set S if
there exists a function f on S so that for every e > 0 there is an integer N such that
|fn (z) − ƒ(z)| < € whenever n > N and z E S.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90b5da61-7c9e-41cf-88f9-128e7ad2af51%2F290b67e6-bebc-47a3-9b4a-216c0dbb6a6a%2F3ogg80i_processed.png&w=3840&q=75)
Transcribed Image Text:4. For z € C, we define the complex exponential by
e²=
=
Σ
n=0
n!
(a) Prove that the above definition makes sense, by showing that the series
converges for every complex number z. Moreover, show that the conver-
gence is uniform5 on every bounded subset of C.
(b) If 2₁, 22 are two complex numbers, prove that e²¹ e²² = e²¹+²². [Hint: Use
the binomial theorem to expand (z₁+z2)”, as well as the formula for the
binomial coefficients.]
5 A sequence of functions {fn(z)}_₁ is said to be uniformly convergent on a set S if
there exists a function f on S so that for every e > 0 there is an integer N such that
|fn (z) − ƒ(z)| < € whenever n > N and z E S.
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